Series Representation of Functions
Welcome to Week 12, the last week of our semester.
We will finish our curriculum content with Lecture 22 on Wednesday, when we look at another important application of the concept of convergence of a sequence, that is the fact that many important functions of mathematics can be represented as infinite power series. A power series in the variable x is like an “infinite polynomial”, it has a constant term, a x term, a x2 term, and so on. To say that the power series represents the function means that at least for some values of x, replacing x with th relevant value in the power series gives an infinite sum that converges to the value of the function at the relevant point. The power series representation may not resemble the definition of the function at all, and in a way this is the key point. Suppose we take a trigonometric function like cos – we know that for any (positive) real number t, cos(t) is the X-coordinate of the point that we would reach by starting at (1,0) and travelling a distance t counter clockwise along the unit circle. This is a definitive descroption of the meaning of cos(t), but it does not help us at all to determine the value of the cosine of 1 or 2 or 25 or any other number (except for some nice rational multiples of pi). But if we had an infinite series converging to cos(t), we could use that to at least approximate the value of cos(t), by truncating the series at some point and taking a finite sum. This principle is used a lot in practice, as are other features of series representation of functions. The concept has great importance in number theory and analysis as well as in modelling, numerical analysis and the study of differential equations.
In Lecture 22, we will look at the Maclaurin series representation of a function, which is the first step in the subject and is a special case of a Taylor series. We won’t prove any theorems, but we will see how our knowledge of differential calculus provides the basic tools needed to derive the coefficients in a power series representation.
Here is (last year’s version of) Lecture 22.
Slides for Lecture 22, without annotation and annotated.
In Lecture 23 on Thursday, we will have a look ahead to the two-hour end of semester exam.
As in recent years, the exam will have six questions, three on calculus and three on algebra. For full marks, all six questions must be answered. The following comments pertain only to the calculus questions.
Each of the three calculus questions has four parts. In terms of content, the three questions correspond to the three chapters of the lecture notes. In Thursday’s lecture, we will have a look at some sample questions and talk a bit about how they could be answered in an exam.
A video version of the exam advice will follow a little later.